Syzygies of Prym and paracanonical curves of genus 8

TL;DR

This paper provides three geometric proofs demonstrating the unexpected failure of the Prym-Green Conjecture for genus 8, challenging previous probabilistic expectations and offering new methods for understanding exceptions in high-divisibility genera.

Contribution

The paper introduces three geometric proofs showing the failure of the Prym-Green Conjecture in genus 8, advancing understanding of syzygies in paracanonical curves.

Findings

Prym-Green Conjecture fails in genus 8

Three geometric proofs established the failure

Methods may explain exceptions in high-divisibility genera

Abstract

By analogy with Green's Conjecture on syzygies of canonical curves, the Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural. The Prym-Green Conjecture is known to hold in odd genus for almost all levels. Probabilistic arguments strongly suggested that the conjecture might fail for level 2 and genus 8 or 16. In this paper, we present three geometric proofs of the surprising failure of the Prym-Green Conjecture in genus 8, hoping that the methods introduced here will shed light on all the exceptions to the Prym-Green Conjecture for genera with high divisibility by 2.